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Theorem lshpne 31980
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lshpne.v  |-  V  =  ( Base `  W
)
lshpne.h  |-  H  =  (LSHyp `  W )
lshpne.w  |-  ( ph  ->  W  e.  LMod )
lshpne.u  |-  ( ph  ->  U  e.  H )
Assertion
Ref Expression
lshpne  |-  ( ph  ->  U  =/=  V )

Proof of Theorem lshpne
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpne.u . . 3  |-  ( ph  ->  U  e.  H )
2 lshpne.w . . . 4  |-  ( ph  ->  W  e.  LMod )
3 lshpne.v . . . . 5  |-  V  =  ( Base `  W
)
4 eqid 2402 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2402 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
6 lshpne.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 31977 . . . 4  |-  ( W  e.  LMod  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W
) `  ( U  u.  { v } ) )  =  V ) ) )
82, 7syl 17 . . 3  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( U  u.  { v } ) )  =  V ) ) )
91, 8mpbid 210 . 2  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  (
( LSpan `  W ) `  ( U  u.  {
v } ) )  =  V ) )
109simp2d 1010 1  |-  ( ph  ->  U  =/=  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754    u. cun 3411   {csn 3971   ` cfv 5568   Basecbs 14839   LModclmod 17830   LSubSpclss 17896   LSpanclspn 17935  LSHypclsh 31973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-lshyp 31975
This theorem is referenced by:  lshpnel  31981  lshpcmp  31986  lkrshp3  32104  lkrshp4  32106  dochshpncl  34384  dochlkr  34385  dochkrshp  34386  dochsatshpb  34452
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